Dislocation-based model for the prediction of the behavior of b.c.c. materials – grain size and strain path effects

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Dislocation-based model for the prediction of the behavior of b.c.c. materials – grain size and strain path

effects

Tales Carvalho Resende, Salima Bouvier, Farid Abed Meraim, Tudor Balan, Simon Serge Sablin

To cite this version:

Tales Carvalho Resende, Salima Bouvier, Farid Abed Meraim, Tudor Balan, Simon Serge Sablin. Dislocation-based model for the prediction of the behavior of b.c.c. materials – grain size and strain path effects. International Journal of Plasticity, Elsevier, 2013, 47, pp.29-48.

�10.1016/j.ijplas.2013.01.003�. �hal-01191889�

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This is an author-deposited version published in: http://sam.ensam.eu Handle ID: .http://hdl.handle.net/10985/9893

To cite this version :

Tales CARVALHO RESENDE, Salima BOUVIER, Farid ABED MERAIM, Tudor BALAN, Simon Serge SABLIN - Dislocation-based model for the prediction of the behavior of b.c.c. materials – grain size and strain path effects - International Journal of Plasticity - Vol. 47, p.29-48 - 2013

Dislocation-based model for the prediction of the behavior of b.c.c. materials – grain size and strain path effects

T. Carvalho Resende

^1,2

, S. Bouvier

¹

, F. Abed-Meraim

³

, T. Balan

³

, S-S. Sablin

²

1 Laboratoire Roberval, UMR CNRS 6253, Université de Technologie Compiègne, Centre de Recherches de Royallieu, BP 20529-60205 Compiègne cedex, France

2 TECHNOCENTRE RENAULT – Vehicle Engineering – Materials Engineering Department – 1 avenue du Golf, 78288 Guyancourt cedex, France

3 Laboratoire d’Etude des Microstructures et de Mécanique des Matériaux, UMR CNRS 7239, Arts et Métiers ParisTech, 4 rue Augustin Fresnel, 57078 Metz cedex 03, France

tales.carvalho.resende@gmail.com, salima.bouvier@utc.fr, tudor.balan@ensam.eu, Farid.ABEDMERAIM@ensam.eu, simon-serge.sablin@renault.com

Corresponding author : Pr Salima BOUVIER, Tel. +33 (0)3 44 23 79 38, Fax +33 (0)3 44 23 49 84

ABSTRACT

Sheet metal forming processes involve multi-axial strain paths. For the numerical simulation of such processes, an appropriate constitutive model that properly describes material behavior at large strain is required. For accurate and time-effective simulations, it is crucial to use plasticity models based on physics, as material macroscopic behavior is closely related to the evolution of the associated microstructures. Accordingly, a large strain work-hardening phenomenological model that incorporates the intragranular microstructure evolution through a dislocation density approach is proposed. The model is defined by a yield criterion and hardening laws that are all grain-size dependent. The classical Hill criterion in which grain- size dependency was introduced is proposed. Hardening laws are given by a combination of kinematic and isotropic contributions that respectively take into account the evolution with strain of cell blocks formed by geometrically necessary boundaries (GNBs) and individual dislocation cells delineated by incidental dislocation boundaries within cell blocks (IDBs). On the one hand, IDBs evolution contribution is described by a modified Rauch et al. isotropic model, which is able to describe work-hardening stagnation and work-softening. On the other hand, GNBs evolution contribution is described by a grain-size dependent tensorial back- stress expression proposed by Aouafi et al. [2007] to describe the plastic anisotropy and Bauschinger effect. Moreover, the proposed model aims to accurately predict steel behavior through an innovative approach by only changing few “simply measurable” microstructure data (e.g. chemical composition, grain size…). The predictive capabilities of the model are assessed for interstitial free (IF) and dual phase (DP) steels with grain sizes varying

respectively in the 8-40 µm and 1-10 µm value range. Different loading paths are analyzed, namely the uniaxial tensile test, reversal simple shear and orthogonal tests.

Keywords: A - Microstructures, A - Strengthening mechanisms, B - Constitutive behavior, A - Yield condition, B - Anisotropic material

1. Introduction

With a view to environmental, economic and safety concerns, car manufacturers need to design lighter and safer vehicles in ever shorter development times. In recent years, several mild steels such as interstitial free (IF) steels and high strength steels (HSS) as dual phase (DP) steels, are increasingly used for sheet metal parts in the automotive industry to reduce mass as they provide excellent drawability. The application of simulation models in sheet metal forming in the automotive industry has proven to be beneficial for reducing tool costs in the design stage and optimizing current processes. The finite element method (FEM) is quite successful to simulate metal forming processes, but the accuracy of such a tool depends on several important factors, such as the ability of the constitutive laws to describe material initial or induced anisotropy and work-hardening, especially after strain-path changes. Some effort is still required to improve the considered behavior models and to identify them in the most optimal way. These behavior models are generally separated in two categories:

phenomenological approaches and physical descriptions. It is well-known that the main advantage of the phenomenological models is the rapidity in computation. Based on a

consistent mathematical description of the observed macroscopic behavior, phenomenological models require fitting of material parameters involved in the constitutive equations on

experimental results (see [Chaboche, 2008] for a general review). Such a process should be repeated for each material characterization. In many practical situations, these models provide effective tools and succeed in describing the material behavior depending on the complexity of the chosen functions. However, no direct physical links exist between the different scales embedded in these models.

Several experimental studies reported that the macroscopic behavior can be attributed to the evolution of the underlying microstructural details, such as dislocation structures (see, e.g., [Hasegawa and Yakou, 1974], [Fernandes and Schmitt, 1983], [Rauch and Schmitt, 1989], [Nesterova et al., 2001ab] among others). More physically based models are developed in order to take into account the consequence of the evolution of the dislocation structures on the macroscopic behavior and specifically when strain-path changes occur (see [Teodosiu and Hu, 1995], [Peeters et al., 2000], [Peeters et al., 2001a], [Yoshida and Uemori, 2002], [Yalcinkaya et al., 2009], among others). Nevertheless, the extensive computing time and resources required by such calculations represent some limitations that may prevent their use in finite element (FE) simulations of industrial forming processes. The goal of the present work is to develop a user-friendly and time-efficient phenomenological model that

incorporates details of the microstructure evolution at the grain scale with a limited number of material parameters. Moreover, particular attention has been focused on keeping a strong physical relevance in describing a wide set of IF and DP steels and realistically reproducing the experimentally observed transients in the macroscopic behavior when strain-path changes occur. Furthermore, because the grain-size strengthening is meant to be the main contribution to work-hardening, an innovative approach that accurately estimates the macroscopic

behavior of different steels by only changing few simply measurable microstructure data (i.e.

chemical composition, volume fraction and grain size) is proposed. This strategy allows avoiding the time-consuming mechanical characterization and identification procedures once a set of fixed physically-based constant parameters is identified.

The outline of the paper is as follows. The macroscopic behavior of IF and DP steels is investigated in Section 2 by means of monotonic and sequential tensile and shear tests, and the underlying microstructural evolutions are analyzed. A dislocation-based constitutive model is derived in Section 3 that describes these evolutions. The parameter identification is addressed in Section 4. Due to the physical significance of several of the model parameters, it is not necessary to identify all of them for each new material, thus reducing significantly the cost of the identification procedure. Section 5 describes in detail the sensitivity of the model to each of these physically-based parameters in order to illustrate the flexibility of the model.

2. Macroscopic behavior and microstructure evolution 2.1. Materials

This study focuses on IF and DP steels with deep drawing quality. IF steels have been designed specifically for automotive applications. These steels are particularly suitable for structural (e.g. longitudinal beams, cross members, B-pillars) and closure applications. They provide an interesting combination in terms of drawability and mechanical strength. These steels have low yield strength and high work-hardening capability. In order to increase both formability and strength, more advanced high strength steels (AHSS), such as dual phase (DP) steels were developed and rapidly became among the most used steels in the automotive industry. Their microstructure typically consists of a soft ferrite phase with dispersed islands of a hard martensite phase.

In the present work, four IF and seven DP steels have been selected and completed by other similar steels from the literature. Their chemical compositions (in % weight) are presented in Table 1 and Table 2, respectively.

Table 1. IF steels chemical composition in % weight.

Materials Reference C Mn P Nb Si Cu Ti N

IF I [Uenishi and

Teodosiu, 2003] 0.0016 0.10 0.005 - N/A - 0.015 0.0018 IF II [Uenishi and

Teodosiu, 2003] 0.0013 0.97 0.005 - 0.014 - 0.014 0.0018 IF III [Uenishi and

Teodosiu, 2003] 0.0015 1.92 0.004 - 0.99 - 0.023 0.0018

IF IV [3DS, 2001] - - - - - - - -

IF V [Cashform, 2001] 0.006 0.16 0.005 - 0.0015 N/A 0.038 -

IF VI 0.007 0.066 0.011 - 0.006 0.002 0.048 -

IF VII 0.007 0.255 0.019 0.0005 0.007 0.007 0.001 -

IF VIII 0.06 0.727 0.013 - 0.074 0.008 0.001 -

IF IX 0.072 1.063 0.015 0.012 0.122 0.003 0.001 -

Table 2. DP steels chemical composition in % weight.

Materials C Mn P Nb Si Cu Ti N

DP I DP 450 0.039 1.241 0.016 0.0008 - 0.005 0.002 -

DP II DP 600 0.115 1.391 0.012 - 0.347 0.007 0.010 0.0047

DP III DP 600 0.071 1.019 0.043 0.0013 - 0.008 0.003 -

DP IV DP 600 0.074 1.849 0.024 0.001 - 0.009 0.002 -

DP V DP 800 0.138 1.902 0.019 0.0012 - 0.011 0.019 -

DP VI DP 800 0.139 1.53 0.017 0.0011 - 0.012 0.003 -

DP VII DP 1000 0.067 2.55 0.014 0.0272 - 0.01 0.02 -

DP VIII DP 1000 0.092 2.485 0.017 0.0195 - 0.007 0.017 -

2.2. Microstructural evolution during monotonic, strain reversal and orthogonal loadings

During deep drawing processes, rolled sheets are subjected to large plastic deformations and multi-axial strain paths, and the entire microstructure evolution can be hardly obtained experimentally. Hence, several studies [Fernandes and Schmitt, 1983], [Rauch and Schmitt, 1989], [Nesterova et al., 2001ab], [Gardey et al., 2005ab] have focused on reproducing some typical deep drawing strain paths through sequences of non proportional mechanical loadings.

Such tests can be monotonic (tensile, shear tests) or series of different monotonic loadings (e.g. reverse or orthogonal strain-path changes…). In this context, the general macroscopic behavior for IF and DP steels is similar to results depicted in [Haddadi et al., 2006].

Explicitly, during strain reversal, IF steels exhibit a strong Bauschinger effect as well as work-hardening stagnation. These characteristic transient effects are followed by resumption of the work-hardening. The latter is related to microstructural evolution given that the initial microstructures are free of organized patterns for annealed materials [Nesterova et al., 2001a].

During monotonic deformation, dislocations arrange themselves into planar polarized dense dislocation boundaries called dislocation walls (DWs). DWs are more or less parallel to the main slip plane and divide the grains into cells [Nesterova et al., 2001a,b], [Franz et al. 2009].

During strain reversal, a stronger Bauschinger effect is observed on DP steels. Two reasons may explain such macroscopic behavior. The presence of the hard phase delays the

dislocation patterning, which makes them less difficult to disintegrate. Moreover, the martensite hard phase introduces additional internal stresses because of its contrasting

behavior compared to the ferrite phase. Slip systems that were active during pre-strain remain active after strain reversal but in the opposite direction. Thus, preformed DWs partially

disintegrate, causing work-hardening stagnation. This transient behavior is more pronounced in IF steels than in DP steels. The formation of new DWs is responsible for work-hardening resumption after stagnation [Rauch and Schmitt, 1989]. After orthogonal loading, contrarily to IF steels, work-softening is absent in DP steels, which is related to the fact that Micro- Bands (MBs) are absent in the latter case [Gardey et al., 2005b]. A simplified summary of the presented microscopic features and their macroscopic counterparts is given in

Table 3.

Table 3. Simplified summary of microscopic features and their macroscopic counterparts for IF and DP steels for low-to-intermediate strain.

Microscopic feature Macroscopic counterpart IF steels DP steels

Formation of DWs Work-hardening Yes Yes

Partial dissolution of polarized DWs Bauschinger effect (BE) Yes Yes Partial dissolution of polarized DWs Work-hardening stagnation Yes Yes

Formation of MBs Work-softening (WS) Yes No

3. Experimental procedure

Uniaxial tensile tests were performed in quasi-static regime at room temperature with a constant strain rate in a tensile machine INSTRON 5582 with an HRD extensometer system monitored by Bluehill®2 software. The obtained results are presented in Figure 1, where the true stress is plotted versus plastic true strain. The elastic strain part is removed using an offset of 0.1% plastic strain.

150 200 250 300 350 400 450 500

0 0.03 0.06 0.09 0.12 0.15 IF VI

IF VII IF VIII IF IX

ε^p

σ [MPa]

(a) 300

400 500 600 700 800 900 1000

0 0.05 0.1 0.15 0.2

DP I DP III DP IV DP V DP VI DP VII DP VIII

ε^p

σ [MPa]

(b)

Figure 1. Uniaxial tensile tests (0° to RD) performed on (a) IF steels and (b) DP steels.

In order to describe the initial yield surface, Lankford coefficients r-values are determined from uniaxial tensile test specimens cut out at different orientations, 0°, 45° and 90° to the

rolling direction of sheet. The r-values are calculated at a certain longitudinal strain (15–

20%) and an average value is used. Planar anisotropy is therefore given by:

0 245 90

m 4

r r r

r ^°+ ^°+ ^°

= , (1)

The experimental results are indicated in Table 4 and Table 5.

Table 4. IF steels grain sizes and standard deviations.

Materials

Y

0

(MPa)

R

m

(MPa)

r

0_°

r

_45°

r

_90°

r

_m D_F

(µm)

DF

σ

(µm)

IF I - - - - - - 40 -

IF II - - - - - - 40 -

IF III - - - - - - 40 -

IF IV 121 - 2.53 1.84 2.72 2.23 25 -

IF V 113 - - - - - 25 -

IF VI 159 286 2.33 1.63 2.52 2.03 22 17

IF VII 200 314 1.82 1.12 1.9 1.49 15 13.5

IF VIII 300 393 0.84 1.01 1.02 0.97 10.5 9

IF IX 342 483 0.64 1.26 0.96 1.03 8 7

Table 5. DP steels ferrite (DF) and martensite (DM) grain sizes. VM stands for martensite volume fraction.

Materials

Y

⁰

(MPa)

R

m

(MPa) ⁰

r

_°

r

_45°

r

_90°

r

_m ^DF

(µm)

DF

σ

(µm)

DM

(µm)

DM

σ

(µm) ^M

V

DP I DP 450 315 460 0.98 0.76 1.34 0.96 9 7.5 9.5 8 0.25

DP II DP 600 285 - - - - - 10 - 3.5 - 0.2

DP III DP 600 409 612 0.74 0.93 0.8 0.85 4 3 3 2.5 0.1

DP IV DP 600 393 593 0.94 0.84 1.39 1 6.5 4.5 2 1.5 0.2

DP V DP 800 530 776 0.77 1 0.86 0.91 3 2.5 2.5 2 0.2

DP VI DP 800 530 802 0.76 0.82 0.98 0.85 3 2 1.5 1 0.15

DP VII DP

1000 700 924 0.68 0.97 0.9 0.88 1.5 1 1.5 1 0.1

DP VIII DP

1000 700 986 0.58 1.09 0.75 0.88 7 5 3.5 3 0.1

Simple and reversal shear tests were performed in the rolling direction (RD) at equivalent conditions in a SERMEES shear assembly implanted in a tensile machine INSTRON 5587.

These tests consist of subjecting flat samples to parallel displacements by fixing the lateral parts (Figure 2). Simple shear tests present the main advantages of reaching high

homogeneous strain levels through simple sample geometry for flat parts. Reversing the load direction during the course of the experiment allows capturing the Bauschinger effect

[Bouvier et al., 2006a]. Monotonic and reverse simple shear tests for different amounts of forward shear strain γT (γT = 2ε12) are performed. For simplicity purposes, the experimental results are reported in the same figures where the model predictions are indicated. An

example of observed behavior is given in Figure 3 for each grade of material. The plots indicate shear stress τ = σ12 versus the plastic part γ of the amount of shear strain. As for the uniaxial tensile test, the elastic part is removed using an offset of 0.1% von Mises equivalent plastic strain.

Figure 2. (a) Schematic representation of the simple shear device; l and h are, respectively, the current length and width of the gauge area and δ_m is the relative displacement of the two lateral faces of the sample; this value is computed using a non contact measurement technique. (b) Deformed specimen with black line indicating the amount of shear strain.

-300 -200 -100 0 100 200 300

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Experimental

IF VIII

γ

τ [MPa]

(a) Strong work-hardening

stagnation

-600 -400 -200 0 200 400 600

-0.3 -0.2 -0.1 0 0.1 0.2

Experimental

γ τ [MPa] DP1000

DP VIII

(b) Strong Bauschinger

transient

Figure 3. Experimental curves of a reversal simple shear test performed at 0° to the RD for (a) IF VIII (DF = 10.5 µm) and (b) DP VIII (DP 1000).

For the orthogonal sequence, the experimental data for IF and DP steels was taken from 3DS report [3DS, 2001]. The simplest orthogonal test consists of the sequence of uniaxial tensile test followed by a simple shear test in the same direction. Further information is given by

[Thuillier and Rauch, 1994], [Bouvier et al., 2006b]. An example of observed behavior is depicted in Figure 4.

100 150 200 250 300 350

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Experimental

σ or τ [MPa]

IF IV

DF=25µm (a)

ε^p or γ Work-softening

200 300 400 500 600 700 800

0 0.1 0.2 0.3 0.4 0.5

Experimental

ε^p or γ

σ or τ [MPa]

DP600 DP II

(b)

Figure 4. Experimental curves of orthogonal loading performed by a sequence of uniaxial tensile test along 0° to the RD followed by simple shear test along 0° to the RD for (a) IF IV (D_F = 25 µm) and (b) DP II (DP 600). Experimental data taken from 3DS report [2001].

All the experimental data used in the present study was pre-processed using home-made Matlab© routines. Stress versus strain experimental curves for uniaxial tensile and shear tests were smoothed in order to filter various possible noises due to the data acquisition.

Optical micrographs for the IF steels and SEM observations for the DP steels of the present work (i.e. IF VI to IX and all DP steels except DP II) were performed. SEM observations were realized with a Cambridge Stereoscan 360 field-emission scanning electron microscope (FE-SEM) chamber. Such observations allowed measuring mean grain sizes and phase volume fractions. Grain sizes were measured using ImageTool software. The procedure consisted of circling approximately 50 grains (for ferrite and martensite) in 4 different images per steel (i.e. two different images from different regions at the surface of the samples and two different images from different regions through sample thickness). Mean grain sizes are calculated by converting the measured cross sectional area of a feature to the diameter of an equal area circle (which is termed as mean grain size). The chosen procedure to measure the grain sizes gives a log-normal distribution in which a large standard deviation was observed.

The martensite volume fraction was calculated by circling martensite regions and dividing their respective area by the total area of one image.

SEM observations revealed that DP steel microstructure is composed by martensite islands distributed in a ferrite matrix. It was found that ferrite mean grain size of the studied IF steels (IF VI to IX) vary in the 8-22 µm value range. The ferrite and martensite mean grain sizes of the studied DP steels (all DP steels except DP II) vary in the 1.5-10 µm value range.

Martensite volume fraction is always under 25%. The obtained values for grain sizes and martensite volume fractions are summarized in

Table 4 and Table 5 for IF and DP steels, respectively.

4. Dislocation-based constitutive modeling

The model is developed in the general framework of cold deformation of metals and the viscous effects on the work-hardening behavior are neglected. At moderately large strains, because the main contribution to the work-hardening is due to the microstructural evolution, only this contribution in connection with the initial crystallographic texture is considered. The texture evolution during work-hardening and its influence on the work-hardening are

neglected. In the modeling, much attention is paid to precisely simulating the behavior of the transient Bauschinger effect, as well as the work-hardening stagnation and work-softening.

After a brief recall of the classical setting of rate-independent anisotropic plasticity, the proposed hardening model is introduced and coupled in the present work to the classical orthotropic [Hill, 1948] criterion. The latter is modified in order to take into account the sensitivity to material mean grain size. Conventional notations as defined in [Haddadi et al., 2006] are used.

4.1. Yield condition and flow rule

The present constitutive model has been developed within the framework of the yield surface concept. Based on the [Hill, 1948] criterion, the yield function f is given by the following equation:

(Y0 ) 0

f =

σ

−Y =

σ

− +R ≤ , (2)

where Y is the yield stress and Y₀ is related to solid-solution and grain-size strengthening. It is worth noting that the initial yield stress is given by Y₀ +R(0). Moreover, work-hardening induced by plastic flow is assumed to be described by a combination of kinematic hardening X and isotropic hardening R.

σ

is the equivalent effective stress and is defined by

: :

σ = T H T, (3)

where T=σ^D−X and σ^D is the Cauchy stress deviator. H is a fourth-order tensor

characterizing the texture anisotropy and has the symmetry properties H_ijkl =H_jikl =H_klji and H_iikl =0.

The plastic flow occurs under the conditions f =0 and

(

^{∂f /∂}σ

)

^:σɺ ^>0. Under the classical normality assumption, the plastic strain rate tensor writes

p f :

λ λ

σ

= ∂ =

∂

D ɺ H T ɺ

σ ^{, (4)}

where

λ

^ɺ is the plastic multiplier and is equal to the equivalent plastic strain rate εɺ.

4.2. Yield criterion

For practical applications, the quadratic [Hill, 1948] criterion is still the most attractive candidate due to its simplicity. In the present study, grain-size dependency of the Hill’48 criterion parameters is proposed. The Voigt notation is used to exploit the symmetry

properties of the forth-order tensor H in order to transform it to a second-order tensor defined by

0 0 0

0 0 0

0 0 0

0 0 0 2 0 0

0 0 0 0 2 0

0 0 0 0 0 2

G H H G

H F H F

G F F G

L M

N

+ − −

 

 

− + −

 

 − − + 

= 

 

 

 

 

 

H ,

(5)

where the components F, G, H, L, M , N remain constant during the deformation if the texture evolution is neglected and are given by

90

0 0

0 90 45

90 0

1 1

( )(2 1)

1

2 ( 1)

F H r

G r

H r G

r r r

N r r

°

°

°

° ° °

° °

 =



 =

 +

 =



+ +

 =

 +



, (6)

r0_°, r_45° and r_90° are the anisotropy coefficients at 0°, 45° and 90° from the rolling direction (RD), respectively. r-values are microstructure dependent and a large ferrite grain size yields high anisotropy coefficient values as given by the following equation [Karlyn et al., 1969], [Renavikar, 2003], [Xu et al., 2006]:

log( _F)

r_α =A_α D +B_α, (7)

where A_α and B_α are constants and D_F is the ferrite mean grain size (Table 6). Subscript

α

stands for the angle between the tensile specimen axis and the rolling direction of the sheet.

The obtained results for 0° and 90° directions are better than for the 45° direction (Figure 5).

Indeed, the linear correlation coefficient in terms of data adjusting is found to be equal to R = 0.99 and R = 0.98, for 0° and 90° respectively, and R = 0.79 for 45° direction (as indicated on Figure 5). This result could be related to the important strain-sensitivity of r_45° [Choi et al., 2006]. The corresponding slope for planar anisotropy is about 2.62, which is in reasonable agreement with literature data [Karlyn et al., 1969], [Xu et al., 2006].

0.5 1 1.5 2 2.5 3

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 y = -3.0703 + 4.0259x R= 0.98856

IF IX IF VIII IF VII

IF VI

log(D

F)

r 0°

IF

IF IV

(a)

1 1.2 1.4 1.6 1.8 2

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 y = -0.16653 + 1.3172x R= 0.79015

log(D

F) r 45°

IF

IF IX

IF VIII

IF VII

IF VI IF IV

(b)

0.5 1 1.5 2 2.5 3

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 y = -2.651 + 3.8296x R= 0.98414

log(D

F)

r 90°

IF

IF IX IF VIII

IF VII IF VI

IF IV

(c)

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 y = -1.5133 + 2.6227x R= 0.96695

IF

IF IX

IF VIII IF VII

IF VI

IF IV

log(D

F)

r m

(d)

Figure 5. Relationship between the ferrite grain size and the anisotropy parameters at (a) 0°, (b) 45°, (c) 90° as well as (d) the planar anisotropy ⁰ 2^{45 90}

m 4

r r r

r ^°+ ^°+ ^°

= for IF steels. R is the linear correlation coefficient.

Table 6. Linear law coefficients for the relationship between the anisotropy parameters and the grain size.

A_α B_α

α

= °0 ^{4.03 -3.07}

α

=45° ^{1.32 -0.17}

α

=90° ^{3.83 -2.65}

For the present DP steels, the linear law for anisotropy parameters with respect to the ferrite average grain size is also attempted. Nonetheless, it is observed that r-values are rather close to 1, which indicates that DP steels are almost isotropic. Thus, in this study, DP steels will be characterized by the von Mises isotropic yield criterion.

4.3. Description of the internal state variables

A phenomenological model that incorporates details of the microstructure evolution at the grain-size scale was developed. The model consists of a combination of isotropic and kinematic contributions and modeling is based on the fact that the strengthening effect of mobile dislocation boundaries (depending on boundary spacing) acts in the same way as stationary grain boundaries (GBs). The key feature of modeling intragranular microstructure evolution consists in considering that cell block spatial dimensions composed of cell block boundaries (GNBs) and cell boundaries (IDBs) continuously refine with stress and strain [Winther and Jensen, 1997]. Hence, the total flow stress can be determined by the sum of IDBs and GNBs strengthening contributions. IDBs are assumed to be randomly oriented cell boundaries, which arise from the mutual trapping of dislocations into low energy

configurations (i.e. dislocations trapped between GNBs and polarity dislocations dissolved from DWs). These boundaries are penetrable to slip and give rise to isotropic hardening.

Thus, the IDBs evolution contribution is described by the Rauch and co-authors isotropic hardening model [Rauch et al., 2007], [Rauch et al., 2011], which is able to describe work- hardening stagnation and work-softening. GNBs/IDBs incompatibility (dislocations glide easily in IDBs but hardly in GNBs) induces back-stresses and gives kinematic hardening at the macroscopic scale [Mughrabi et al., 1986], [Wilson and Bate, 1996]. Phenomena such as the Bauschinger effect are described through the back-stress. A grain-size dependent back- stress description was proposed by Sinclair et al. [2006]. A tensorial expression for the back- stress was further proposed by Aouafi et al. [2007] in order to describe the plastic anisotropy.

4.3.1 Isotropic work-hardening

The Taylor formulation [Taylor, 1934] offers an adequate framework to describe plasticity related to the dislocation densityρ. The yield stress can be written as

(

⁰

) (

⁰

)

I I I I I I

I I

Y =∑V Y +R =∑V Y +M Gbα ρ , (8)

and

( )

⁰

( )

⁰

I I

R ⁼M Gb

α ρ

. (9)

I represents the different phases in the material (e.g. ferrite, martensite…) with V_I their respective volume fraction. In this work, for DP steels, I =F M, ; where F stands for

ferrite/ferrite interface and M for ferrite/martensite interface. G stands for the shear modulus and b for the magnitude of the Burgers vector and their values are set to be G=80 GPa and

2.5 x 10 m-10

b= , respectively. M is the Taylor factor and is assumed to be equal to the average Taylor factor M . In this work, we assume that the studied steels have perfect γ -

fiber. Hence, we assume that M is equal to 3.14 [Murakami, 2008].

α

represents the dislocation interaction and is taken equal to 0.4 [Feaugas, 1999]. For IF steels, the initial dislocation density is set to be equal to

ρ

⁰ =10 m^{12 -2}. In DP steels, the dislocation density in the ferrite/martensite interface is higher than within the ferrite grain, their initial dislocation density is equal to

ρ

⁰ =10 m^{13 -2} [Gardey et al., 2005ab]. Y_0n is estimated through grain size and chemical composition using the following combination of Hall-Petch and Pickering [1992] relations:

( )

[ ]

^0.54

[ ] [ ] [ ] [ ] [ ] [ ]

0_n ^{HP n} 42 Mn 750 P 60 Si 80 Cu 80 Ti 2400 Nb 5000 N

F

Y k

= D + + + + + + + _{, (10)}

where [.] designates the weight percent concentration of the solutes. k_HP is the Hall-Petch coefficient (k_HP ≈15 18 MPa mm− for low-carbon steels [Gladman, 1997]). In the present work, we opted for an intermediate value for IF steels: k_HP =16.5 MPa mm. In DP steels,

20 MPa mm

kHP = [Morrison, 1966].

The dislocation density evolution with strain is based on the equations proposed in the [Rauch et al. 2011] model, which consists in describing the intragranular evolution of dislocation structure by splitting the total dislocation density in three components:

/ / /

I F I R I L I

ρ

=

ρ

+

ρ

+

ρ

, where

ρ

_F/I,

ρ

_R/I,

ρ

_L/I are associated with forward, reverse and latent substructures, respectively. It is important to stand out that the total dislocation density (ρ) is the only one that can be experimentally measured and consequently has a physical meaning.

The other terms (

ρ

_F/I,

ρ

_R/I,

ρ

_L/I) are phenomenological distributions of this total dislocation density. These terms are taken over from their respective hardening counterparts.

Initially, and throughout monotonic loading,

ρ

R/I =

ρ

L/I = 0. When a reverse strain-path change occurs, a fraction p of the total dislocation density cumulated during pre-strain is associated with dislocations that move along the opposite slip direction of previously activated slip systems. This “reverse” type dislocation density corresponds to

ρ

R/I and the remaining part of

Ipre

ρ

accounts as “forward”: ρ_{R I}⁰_/ = p

Ipre

ρ

; ρ_{F I}⁰_/ =

Ipre

ρ

- ρ_{R I}⁰_/ . After an orthogonal strain-path change (e.g., tensile test followed by shear test in the same direction), the entire dislocation density becomes of “latent” type: ρ_{L I}⁰_/ =

Ipre

ρ

; ρ_{F I}⁰_/ = 0. Here, “pre” designates values at the end of the pre-strain stage and “0” designates values at the beginning of the subsequent stage.

These definitions are sufficient for the simulation of the one-dimensional two-step strain paths typical of mechanical tests used for parameter identification. Along a general strain-path change, the total dislocation density can be decomposed into its three components according to the following more general rule:

( )

0 2

/

0 2

/

0 0 0

/ / /

1

pre

pre

pre

R I I

L I I

F I I R I L I

ρ

p

θ ρ

ρ θ ρ

ρ ρ ρ ρ

=

= −

= − −

, (11)

where θ is the classical Schmitt factor

pre:

θ =N N, (12)

and N = D^p D^p is the direction of the plastic strain rate tensor.

The evolution laws for the “forward”, “reverse” and “latent” components of the isotropic hardening internal variable are given by

/

/

/

3

/ 1 / 2 /

3 /

/ 1 /

/ 2 /

F I

R I

pre

L I

I

F I F I F I

I

I R I

R I F I

I I

L I L I

H ε M k k k ε

bD

H ε M k k ε

bD

H ε Mk ε

ρ

ρ

ρ

ρ ρ ρ

ρ ρ ρ

ρ

ρ ρ

 

=  − + 

 

 

= −  + 

 

= −

ɺ ɺ

ɺ

ɺ ɺ

ɺ

ɺ ɺ

ɺ

=

=

=

, (13)

where k₁, k₂ and k_3I are related to the athermal work-hardening limit and the dynamic recovery, respectively. k_3F =k₃, the value taken for the IF steels and k_3M =8MV_M, M is the Taylor factor and VM is the volume fraction of the martensite phase. k₃ is a geometric factor related to the proportion of dislocations arriving at the boundaries. k₁, k₂ and k₃ are fitting parameters and are set to be constant. The description of the work-hardening stagnation and softening is correlated to the k k_{1 2} combination.

4.3.2. Kinematic work-hardening

The kinematic contribution for a monotonic loading is given by [Sinclair et al., 2006]:

1 e ⁰

λ ε bn 0

F

X M Gbn D

 − 

=  − 

 

, (14)

where MGbn D_{0 F} stands for the saturation value of the kinematic contribution and

λ

bn₀ is the saturation rate of the kinematic contribution.

λ

is the mean DWs spacing, which

decreases with strain. Nonetheless, for the sake of simplicity, because this parameter quickly reaches its final value (after approximately 3%-4% plastic strain [Uenishi, 2003]), we assume it as a constant. n₀ is the “maximum number of dislocations” that can be stored at a boundary on a given slip band (or slip line).

λ

_and n₀ are fitting parameters.

Because Eq. (14) is a phenomenological description in which polycrystals are characterized with what is observed in single crystals, n₀ should rather be considered as a parameter that introduces the concept of internal stress saturation. By differentiating Eq. (14) we obtain:

0

0 F

X M Gbn X ε

bn D

λ

^{ }

=  − 

 

ɺ ɺ, (15)

During uniaxial deformation, the kinematic hardening introduced in Eq. (15) coincides with the phenomenological non-linear evolution law proposed by Armstrong and Frederick [1966].

Hence, its tensorial expression is given by (see, e.g. Aouafi et al. [2007]):

( )

X sat

C X ε

=

Xɺ N - X ɺ_{, (16)}

where _{sat 0}

F

X = M Gbn

D and _X

0

C bn

= λ .

It is worth noting that contrarily to isotropic hardening, the kinematic hardening expression remains unchanged for DP steels.

5. Numerical procedure 5.1. Model identification

The phenomenological model that incorporates details of the microstructure evolution at the grain-size scale presented previously was implemented in MATLAB. The identification of the set of fixed physically-based constant parameters (i.e. k₁, k₂, k₃, λ and n₀) was carried out using the fminsearch routine, which attempts to find a constrained minimum of a scalar function of several variables starting from an initial parameter set. The set of experimental tests used for the material parameter identification is composed of six mechanical tests: a uniaxial tensile test, a simple shear test, a reverse simple shear (i.e. Bauschinger) loading at three amounts of shear pre-strain (10%, 20% and 30%) and an orthogonal loading (10%

tensile pre-strain followed by a simple shear test). It is noteworthy that identification should be done on coarse grain size materials because the effect on the grain boundary region is negligible [Pipard et al., 2009]. The obtained results for IF and DP steels are shown in Figure 6. The evolution of dislocation densities for IF IV and DP II during reverse and orthogonal loading is shown in Figure 7 and Figure 8, respectively. Note the stagnation and the decrease of the total dislocation density during reverse and orthogonal loadings for IF IV. The fitting parameter values for IF and DP steels are presented in Table 7 and Table 8, respectively.

For industrial applications, the use of orthogonal tests increases the identification time. Hence, the previous identification procedure was done again for IF IV using only two mechanical tests: uniaxial tensile test and reverse loading with 30% of shear pre-strain. The obtained fitted parameter values are reasonably close to those of the complete identification procedure, which means that a reduced experimental basis can be used without important loss in

accuracy. The corresponding material parameter values are reported in Table 9.

-300 -200 -100 0 100 200 300 400 500

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Experimental

Proposed model

ε^p or γ

σ or τ [MPa]

DF=25µm

IF IV

(a) -400

-200 0 200 400 600 800 1000

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Experimental

Proposed model

ε^p or γ

σ or τ [MPa]

DP600 DP II

(b)

Figure 6. Results of the Advanced Microstructural model identification procedure for (a) IF IV steel and (b) DP II steel using 6 mechanical tests I.

0 2 10⁷ 4 10⁷ 6 10⁷ 8 10⁷ 1 10⁸ 1.2 10⁸

0 0.2 0.4 0.6 0.8 1

ρF

ρR

ρL

ρ

Dislocation density (mm-2 )

(a)

γ

0 2 10⁷ 4 10⁷ 6 10⁷ 8 10⁷ 1 10⁸ 1.2 10⁸

0 0.2 0.4 0.6 0.8 1

ρ_F ρ_R ρ_L ρ

Dislocation density (mm-2 )

(b)

ε^p or γ

Figure 7. IF steel (IF IV) dislocation density evolution related to (a) reverse and (b)

orthogonal loading. Note the stagnation in case (a) and the softening in case (b) of the total dislocation density.

0 2 10⁸ 4 10⁸ 6 10⁸ 8 10⁸ 1 10⁹ 1.2 10⁹ 1.4 10⁹

0 0.2 0.4 0.6 0.8 1

ρF F/M

ρR F/M

ρL F/M

ρ^F/M

Dislocation density (mm-2 )

(a)

γ

0 2 10⁸ 4 10⁸ 6 10⁸ 8 10⁸ 1 10⁹ 1.2 10⁹ 1.4 10⁹

0 0.1 0.2 0.3 0.4 0.5 0.6

ρF F/M

ρ_R^F/M ρ_L^F/M ρ^F/M Dislocation density (mm-2 )

(b)

ε^p or γ

Figure 8. DP steel (DP II) ferrite/martensite boundary dislocation density evolution related to (a) reverse and (b) orthogonal loading. Note that stagnation and no softening of the total ferrite/martensite boundary dislocation density occur in case (a) and (b), respectively.

Table 7. Advanced Microstructural model fitted parameters for IF steel IV with six mechanical tests.

k1 (nm^-1) k₂ k₃

λ

^{(nm) n}₀

p

0.031 2.7 0.1 415 10.3 0.83

Table 8. Advanced Microstructural model fitted parameters for DP steel II with six mechanical tests.

k1 (nm^-1) k₂ k₃

λ

^{(nm) n}₀

p

0.033 3.1 0.1 61 6.2 0.81

Table 9. Advanced Microstructural model fitted parameters for IF steel IV with two mechanical tests (tensile and reverse loadings).

k1 (nm^-1) k₂ k₃

λ

^{(nm) n}₀

p

0.035 2.6 0.1 360 11.1 0.86

5.2. Grain size effect simulation

Many recent works have been focusing on grain size effects on the overall plastic behaviour through more “homogenized” material models [Chia et al., 2005], [Sinclair et al., 2006], [Pipard et al., 2009]. These “homogenized” models have proven to quantitatively estimate the grain size effect through relatively easy equations (e.g. Kocks-Mecking-Estrin theory).

Because identification procedures are known to be time-consuming, car manufacturers try to avoid them whenever possible. Hence, the aim of this section is to assess an approach that allows avoiding identification procedures. This approach consists in accurately estimating

steel behaviour by only changing few microstructure data, such as ferrite D_F and martensite DM mean grain sizes, martensite volume fraction VM and the chemical composition in Eqs.

(10), (13) and (14). Ferrite and martensite mean grain sizes and martensite volume fraction affect the yield stress and work-hardening while the chemical composition only affects the initial yield stress. The proposed predictive approach is obtained through a set of fixed physically-based constant parameters (k₁, k₂, k₃,

λ

, n₀) that are identified only once. This means that the same material parameters (those identified only once for IF and DP steels) are subsequently used to simulate the response of other IF and DP steels. The simulated curves are able to quantitatively predict the effect of grain size on work-hardening for IF and DP steels as depicted in Figure 9 and Figure 10 to Figure 13, respectively.

-300 -200 -100 0 100 200 300 400 500

-0.15 0 0.15 0.3 0.45 0.6 Experimental

Proposed model

ε^p or γ

σ or τ [MPa]

IF VI

(a)

DF=22µm

-375 -250 -125 0 125 250 375 500

-0.3 0 0.3 0.6 0.9

Experimental Proposed model

ε^p or γ

σ or τ [MPa]

IF VII

(b)

DF=15µm

-400 -200 0 200 400 600

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Experimental

Proposed model

ε^p or γ

σ or τ [MPa]

IF VIII

(c)

DF=10.5µm

-300 -150 0 150 300 450 600 750

-0.3 -0.15 0 0.15 0.3

Experimental Proposed model

ε^p or γ

σ or τ [MPa]

IF IX

(d)

DF=8µm

Figure 9. Estimated (obtained without any identification procedure) work-hardening behavior of IF VI to IX (grain size varies from 22 µm to 8 µm) through a set of fixed physically-based constant parameters (k₁, k₂, k₃,

λ

, n₀) for IF steels. Only the mean grain size and chemical composition are modified.